How to get podcasts on the new Apple TV

The new Apple TV, perplexingly, shipped without a built-in Podcasts app. Perplexingly because the icon for the podcasts app shows up in pre-release material, which seems to indicate it's been planned but, for some reason, just couldn't ship on time. Regardless, you have podcasts, you want them on your Apple TV, and you want them now. Here are some options.